10 A Cycle in a Residual Networkactual meaning of sending flow in cycle s-t-1-s (Figure 4)a different way to send flow from s to tflow along t-1-s  reduction of flow in s-1-t(-8,3)(-2, 3)(6, 4)(8, 2)(2, 4)Figure 4. The Residual Network Corresponding to Figure 2st1Figure 5. The New Residual Network After Adding a Flow of 3 units in cycle s-t-1-sst1(6, 1)(8, 5)(2, 7)(-6, 3)st13Figure 6. Flows on arcs10

11 Potentials of Nodes and the Corresponding Reduced Costs of Arcsfirst the ideacall any arbitrary set of numbers {i}, one for a each node, a set of potentials of nodesdefine the reduced cost with respect to this set of potentials {i} byinteresting properties for this set of reduced costs11

12 Potentials of Nodes and the Corresponding Reduced Costs of ArcsC(P) be the total cost of path PC(P) be the total reduced cost of path Pi = potential of node iP = a path from node s to node tthen C(P) = C(P)  s + t12

13 Potentials of Nodes and the Corresponding Reduced Costs of ArcsC(P) be the total cost of path PC(P) be the total reduced cost of path Pi = potential of node iP = a path from node s to node t, thenC(P) = C(P)  s + ta shortest path from s to t for C() is also a shortest path from s to t for C()123456871 = 32 = -53 = 24 = 65 = -36 = 5C(P)C(P)1234561018-2131413

18 Ideas for Theorem 1a cycle in a residual network: a different way to send flow across a networkexisting of a negative cycle  existence of another flow pattern with lower costno negative cycle  no other flow pattern with lower cost18